| 1. |
- Englis, Miroslav, et al.
(författare)
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On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions
- 2004
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Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; 570, s. 89-129
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Tidskriftsartikel (refereegranskat)abstract
- In an earlier work, J.-L. Lions produced a means for creating reproducing formulas for the space of harmonic functions with Sobolev boundary values on a domain in real Euclidean space. The present authors give a new proof of this result and also generalize the Lions formula to handle spaces of functions that are annihilated by an elliptic operator. The method of constructing the reproducing kernels seems to be based on the old paradigm of Aronszajn and Bergman. It is interesting to note that this model---that the kernel should take the form $$ K(x,y)=\sum_j e_j(x)·e_j(y) $$ for a suitable orthonormal basis $\{e_j\}$---goes back to the thesis of Bochner. That thesis well predates the early work of Bergman and Szegö.
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| 2. |
- Hoibakk, Ralph, et al.
(författare)
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A New Development of the Classical Single Ladder Problem via Converting the Ladder to a Staircase
- 2021
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Ingår i: Mathematics. - : MDPI. - 2227-7390. ; 9:4
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Tidskriftsartikel (refereegranskat)abstract
- Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations are given and examples of new results are also given. In particular, we derive a number of integer SLPs with congruent ladders, where a set of rectangular boxes with integer sides constitutes a staircase along a common ladder. Finally, the case with a regular staircase along a given ladder is investigated and illustrated with concrete examples.
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| 3. |
- Høibakk, Ralph, et al.
(författare)
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A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation
- 2020
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Ingår i: Mathematics. - : MDPI. - 2227-7390. ; 8:2, s. 1-21
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Tidskriftsartikel (refereegranskat)abstract
- The aim is to put new light on the single ladder problem (SLP). Some new methods for finding complete integer solutions to the corresponding quartic equation z(4) - 2Lz(3) + ( L-2 - a(2) - b(2))z(2) + 2La(2)z - L(2)a(2) = 0 are developed. For the case L >= L-min, these methods imply a complete parametric representation for integer solutions of SLP in the first quadrant. Some corresponding (less complete) results for the case L > L-min are also pointed out.
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| 4. |
- Høibakk, Ralph, et al.
(författare)
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On some power means and their geometric constructions
- 2018
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Ingår i: Mathematica Æterna. - : Hilaris Ltd. - 1314-3336 .- 1314-3344. ; 8:3, s. 139-158
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Tidskriftsartikel (refereegranskat)abstract
- The main aim of this paper is to further develop the recently initiatedresearch concerning geometric construction of some power means wherethe variables are appearing as line segments. It will be demonstratedthat the arithmetic mean, the harmonic mean and the quadratic meancan be constructed for any number of variables and that all power meanswhere the number of variables are n = 2m, m 1 2 N for all powersk = 2?q and k = 2q; q 2 N can be geometrically constructed.
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| 5. |
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| 6. |
- Lions, Jacque-Louis, et al.
(författare)
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Reiterated homogenization of monotone operators
- 2000
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Ingår i: Comptes rendus de l'Académie des sciences. Série 1, Mathématique. - 0764-4442 .- 1778-3577. ; 330:8, s. 675-680
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Tidskriftsartikel (refereegranskat)abstract
- In this Note we study reiterated homogenization of nonlinear equations of the form −div(a(x/,x/2,Du))=f, where a is periodic in the first two arguments and monotone in the third. We state that u converges weakly in W1,p(Ω) (and even in some multiscale sense), as →0, to the solution u0 of a limit problem. Moreover, we give an explicit expression for the limit problem.
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| 7. |
- Lions, Jacques-Louis, et al.
(författare)
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Reiterated homogenization of nonlinear monotone operators
- 2001
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Ingår i: Chinese Annals of Mathematics. Series B. - 0252-9599 .- 1860-6261. ; 22B:1, s. 1-12
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, the authors study reiterated homogenization of nonlinear equations of the form -div(a(x, x/ε, x/ε2,Duε)) = f, where a is periodic in the first two arguments and monotone in the third. It is proved that uε converges weakly in W1,p(Ω) (and even in some multiscale sense), as ε → 0 to the solution u0 of a limit problem. Moreover, an explicit expression for the limit problem is given. The main results were also stated in [15]. This article presents the complete proofs of these results
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| 8. |
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| 9. |
- Lukkassen, Dag, et al.
(författare)
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Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations
- 2012
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Ingår i: Mathematical methods in the applied sciences. - : Wiley. - 0170-4214 .- 1099-1476. ; 35:11, s. 1300-1311
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Tidskriftsartikel (refereegranskat)abstract
- We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces L p,Ψ(ℝ n,w), defined by an almost increasing function Ψ(r) and radial type weights. We obtain sufficient conditions, in terms of numerical characteristics, that is, index numbers of the weight functions and the function Ψ. In relation with the wide usage of singular integral equations in applications, we show how the solvability of such equations in the generalized Morrey spaces depends on the main characteristics of the space, which allows to better control both the singularities and regularity of solutions
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| 10. |
- Lukkassen, Dag, et al.
(författare)
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Hardy Type Operators in Local Vanishing Morrey Spaces on Fractal Sets
- 2015
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Ingår i: Fractional Calculus and Applied Analysis. - : Springer Science and Business Media LLC. - 1311-0454 .- 1314-2224. ; 18:5, s. 1252-1276
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Tidskriftsartikel (refereegranskat)abstract
- We obtain two-weighted estimates for the Hardy type operators fromlocal generalized Morrey spaces Lp,ϕloc (X,w1) defined on an arbitrary underlyingquasi-metric measure space (X, μ, ) with the growth condition, toLq,ψloc (X,w2), where w1 = w1[(x, x0)], x0 ∈ X is a weight of radial type,while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneouslytreat a similar boundedness V Lp,ϕloc (X,w1) → V Lq,ψloc (X,w2) forvanishing Morrey spaces. We obtain sufficient conditions for such estimatesin terms of some integral inequalities imposed on ϕ, ψ and w1.w2. We alsospecially treat the one weight case where w2(x) is also of radial type. Wedo not impose doubling condition on the measure μ, but base our result onthe growth condition.The obtained results show the explicit dependence of the mapping propertiesof the Hardy type operators on the fractional dimension of the set(X, μ, ). An application to spherical Hardy type operators is also given.
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